Learning Subgroup Relations Using Siamese Graph Neural Networks

This paper addresses the problem of determining subgroup relations between finite groups using geometric deep learning. The author proposes a Siamese GNN architecture, which processes two groups simultaneously through a shared encoder. This ensures that both groups are mapped into a common latent space, allowing the model to learn and compare their structural properties consistently. The framework further enhances these learned graph embeddings by concatenating them with explicit algebraic features, such as the ratio and remainder of the group orders, before passing them to a final classifier.

The proposed Siamese GNN demonstrates high effectiveness in predicting subgroup relations. By integrating structural information derived from Cayley graphs with domain-specific algebraic features, the model achieved a 95.9% accuracy rate on an independent test set. The ablation study highlights that combining graph-level embeddings with the remainder of the group orders provides the most informative representation for the classifier, outperforming models that rely solely on graph structure or different combinations of algebraic features.

Computational group theory traditionally relies on exact symbolic algorithms to analyze group structures. This research demonstrates that deep learning models can serve as a powerful, complementary tool for these tasks. By bridging the gap between algebraic structures and graph-based machine learning, the study provides a flexible, modular framework that can potentially be extended to other classification problems in computational algebra.